Packing vector spaces into vector spaces

A partial t-spread in Fq is a collection of t-dimensional subspaces with trivial intersection such that each non-zero vector is covered at most once. How many t-dimensional subspaces can be packed into Fq , i.e., what is the maximum cardinality of a partial t-spread? An upper bound, given by Drake and Freeman, survived more than forty years without any improvement. At the end of 2015, the upper bounds started to crumble. Here, we give self-contained elementary proofs of the current (unpublished) state of the art—inviting the reader to pursue the newly opened path to improved upper bounds for partial t-spreads.